INSTYTUT PODSTAWOWYCH PROBLEMÓW TECHNIKI STOSOWANEJ ARCHIVES DE MECANIQUE APPLIQUEE 4 XVI. V O L. XVI WARSZAWA, 1964 No. 4

Transkrypt

1 P O L S K A A K A D E M I A N A U K INSTYTUT PODSTAWOWYCH PROLEMÓW TECHNIKI A R C H I W U M M E C H A N I K I STOSOWANEJ ARCHIVES DE MECANIQUE APPLIQUEE 4 XVI V O L XVI WARSZAWA, 1964 No 4 P A Ń S T W O W E W Y D A W N I C T W O N A U K O W E

2 ARCHIWUM MECHANIKI STOSOWANEJ 4, 16 (1964) MIXED OUNDARY-VALUE PROLEMS IN HEAT CONDUCTION W NOWACKI (WARSZAWA) 1 Inroducion Ref [1], by he same auhor, is devoed o he problem of mixed boundary condiions for a saionary hea flow in a solid The mehod presened here will now be generalized o non-homogeneous mixed boundary condiions and o problems of non-saionary hea flow The mehod will be described in a manner somewha differen from [1], differen fundamenal sysems being used A new soluion will also be given for a body wih slis Le us consider a simply conneced body, bounded by he surface S Le his surface be composed of hree regular surfaces S u S 2, S 3 wih common edges a and P (Fig 1) Le ime-variable hea sources W(P, ) Pebe locaed in he body, which FIG 1 is heaed on he surfaces S;(j = 1, 2, 3) The emperaure field hus generaed T(P, ) is described by he hea equaion (11) xv*t(p, )-f(p, ) = -M(P, ), V* = -^ + + ^ In his equaion, x = 1/QC is a maerial consan, where X is he coefficien of hea conducion g he densiy and c he specific hea The funcion M(P, ) expresses he inensiy of he hea sources We have M{P, i) = W(P, ), where W(P, ) is he quaniy of hea produced per uni volume and ime and T is he ime deri-

3 866 W Nowacki vaive of he emperaure f = 8T/8 I is assumed ha T(P, ) saisfies he iniial condiion (12) T(P,0)=/(P), Pe, and he boundary condiions!, 0 = PiCRi, ) on he surface S l R ± e S u (13) -->" 2 ' *' = fi(r 2, ) on he surface S i9 R 2 es 2, on T(R 3, i) = (p z {R 3, ) on he surface S a, i? 3 eą Le us denoe, in general, T(Ri, ) = piri, f), T ^' *' = y>i(r,, ), i = 1, 2, 3 Le us observe ha he funcions cp x, y> 2, <p 3 prescribed on he surfaces S u S 2, S 3 are known, while y l q> 2 and y 3 are unknown funcions on he same surfaces 2 Firs Soluion Mehod Le us consider a "fundamenal sysem" in which Green's funcion G(P, Q, ) of he problem saed in he firs secion will be deermined Le us assume ha he surfaces Ą and S 2 are hermally insulaed, he surface S 3 being kep a zero emperaure Le us observe also ha i is impossible o have an insulaion on he surface S 3, because hen, hea exchange across he surface S being impossible, Green's funcions would have no sense Le us deermine- Green's funcion G(P, Q, ) saisfying he hea equaion (21) x V 2 G (P, Q, 0 - G(P, Q,i)=- d(p - Q) 3(0, P, Q 6, wih he homogeneous iniial condiion (2-2) G(P, Q, 0) = 0, and he homogeneous boundary condiions 8G ( R i>q>ą o = On he surface S l R l e Ą, (23) G(R 3, 6, 0 = 0 on he surface S, R s es 3 On he righ-hand side of (21), Dirac funcion is involved o express he acion of an insananeous poin hea a Q, where

4 Mixed boundary-value problems in hea conducion 867 Le us perform on Eqs (11) and he boundary condiions (13), one-sided Laplace ransformaion as defined by equaion (24) F(P,p) = j F(P, )e-"d, p>0 o I is assumed ha he acion of he hea sources and he surface heaing on S sars a he momen = 0 + As a resul, we obain Eq (1) in he ransformed form (25) xv*f(p,p)-[pf(p,p)-np,q)]=-m(p,p), T(P, 0) = The Laplace ransformaion performed on he boundary condiions (13) yields (26) R 3 es 3 For Eq (21) and he boundary condiions (23) we proceed in a similar way We obain (27) xv*g(p, Q,p)- P G(P, Q,p) = -5(P-Q), Le us make use of Green's formula (29) S by subsiuing appropriae values from Eqs (25)-(28) earing in mind ha we obain from (29) / T(P,p)6(P- Q)d P = T(Q,p), (210) T{Q,p) = ffjf(p)g(p,q,p)d P + }JjG(P,Q,p)M(P,p)d P -j- H j j yi^ri, p) G(J? l5 Q, p)ds Rl + x I J y^r?,,p) G(i? 2, Q, p)ds Rl s, lg(r 3,Q,p) ds dn

5 868 W Nowacki Le us observe ha he inegrals on he righ-hand side of his equaion can be deermined, excep for // ip 1 GdS Rl Si For, he funcions y>, <p 3, f and M are prescribed and G has already been found from (21) Equaion (213) may be represened in he form (211) f(q, p) = T a (Q, p) + x JJ Wx(Ri, P) G(R X, Q, p)ds Ri Si because in he fundamenal sysem described he emperaure T 0 (Q, f) can be reaed as a soluion of he equaion (212) K V 2 r 0 (P, 0 - f(p, )=- M(P, ), Pe, wih he iniial condiion T(P, 0) = f(p) and he boundary condiions (213) STpjR^) "a = u > K i e <Ji> a dur 2,) T 0 (R ) = (p 3 (R 3, ), i? If one-sided Laplace ransformaion is performed on Eq (212) and he boundary condiions (213) and Green's formula (214) hen, afer some simple ransformaions, we obain (215) T 0 (Q,P) - JfJf(P)G(P, Q,p)d P + JfjM(P,p)G(P, Q,p)d P sj J + x J J %CR 2J p) G(R 2, Q,p)dS R2 - K fj lp 3 (R 3,p) s s Performing on he expression (215) he inverse Laplace ransformaion, we have (216) T 0 (Q,) = f f j f(p)g(p,q,)d P + f dr j f j M(P,T)G(P,Q,-r)d P f f j f OO + x J dx j J V2(R 2, T) G(R Q, - x) ds Rl 0 S, 0 S, Le us consider he expression (211) on which he inverse Laplace ransformaion will be performed (217) T{Q, i) - T 0 (Q, ) + jdjj Vl(R l r) G(R l Q, - r)ds Rl,

6 Mixed boundary-value problems in hea conducion 869 ip l (R l T) is an unknown funcion on he surface Ą Le us make he poin Q,e end o he poin R[, e S u on he surface S x Remembering ha T(R[, i) = 9>i(Ki> 0 is he boundary condiion on Ą, we obain from (217) {218) <Pi(Ri f) = T 0 (R{, i) + H J dx JJ ip^r^ r) G(,R 1} R[, r) ds Rl o s, In his inegral equaion, he only unknown funcion is y x (R x, ) On deermining i, we obain he emperaure T(Q, ) from (217) Le us observe ha he Green's funcion G (P, Q, ) may be expressed by anoher Green's funcion defined in he following way: Le us consider he funcion K(P, R[, ) which saisfies he hea equaion (219) n V 2 K(P, R[, f) - K(P, R[, ) == 0, wih he iniial condiion K(P, R' ly i) = 0 and he boundary condiions (220) 8K(R 2,R{,) _ Q R S K(R 3, R l ) = 0, R 3 e S 3 The firs of he condiions (220) ells us ha a concenraed and insananeous hea flow akes place a he poin R Xi e S x of hermal insulaion, and o s, If now Green's formula (29) is applied o he funcions K and G, we obain he relaion (221) JSJK(P,R' 1,p)d(P-Q)d P == xfjg(r 1,Q,p)5(R 1 -Ri)dS Rl Hence (222) Making use of his relaion in (218), we obain (223) Vl {Ri\ 0 = T 0 (R[, )+Jd JJ %(*!, TiKiR, R[, - r)ds Rl o s, Le us consider he paricular case in which he emperaure field varies harmonically wih ime Then, wih (224) M(P,) = L(P)e ia ", T(P, ) = 0(P, co)e ia " Eq (11) akes he form (225) x V 2 & (P, co) - ico 6 (P, co)= -L(P) Si

7 870 W Nowacki On inroducing he noaions cp^r^ i) = 0 1 (R 1 )e im, f 2 (R 2, i) = fwe 1 ", cp z {R z, ) he boundary condiions (13) can be wrien hus:,w) = 0 (R ), R es, 1 1 (226) T Le us denoe Green's funcion by 3, a>) = & 3 (R 3 ), and solve (21) wih he boundary condiions (23) Similarly, le us solve (212) wih he boundary condiions (213) We inroduce he noaion T Q (P, ) = Q 0 (P, co)e im Making use now of he Green's formula (29) for he funcions F, and 6, we find he following expression for he ampliude 6(Q, co) (227) 6(Q, co) = e o (Q, «) + «// Wi)AĄ, Q, co)ds Ri, where Si (228) d o (Q, OJ) = /// L(P)F(P, Q, co)d P + xjj^(r^fir,, Q, co)ds R s, The funcion W ± (R^) will be deermined from he boundary condiion (229) 0 O (*;, co) + xjj W 1 (R^r( R l3 Rl co)ds Rl = 0^), R u R^eS 1 The funcion ^(JRJ having now been deermined, we can find 9 0 (Q, co) from he inegral expression (227) I should be observed ha a soluion analogous o (227)- (229) can be obained for he Helmholz equaion (230) W 2 F+X 2 F=-L, wih he boundary condiions (226) for he funcion F In he soluion of Eq (225), we should only replace & wih F and co wih +z'p Reurning now o he problem of hea conducion, le us observe ha for m -* 0 ha is, for a hea wave wih infinie period he problem ends o ha of saionary hea flow Equaion (11) becomes Poisson's equaion Denoing he emperaure in his sae by T(P), he inensiy of hea sources by M(P), Green's funcion by G(P, Q) ec, we obain for he emperaure equaion (231) T(Q) = T 0 (Q) + KJJ Vl (iy G(R l Q)dS Rl, SI

8 Mixed boundary-value problems in hea conducion 871 where (232) T o (0 = jff M(P)G(P, Q)d P + JJ ^(R,) G(R 2, Q)dS Rl The unknown funcion y) 3 (R^) will be obained from he boundary condiion ( Pl (Ri)=T 0 (R' 1 ),R 1 es 1 (233) T 0 (Rd + x SI yi(*i)gcri R'i,)dS Rl = Vl (R[) Si The above mehod for deermining he emperaure disribuion wih mixed boundary condiions is useful if Green's funcion G (P, Q, ) can be obained in he fundamenal sysem assumed For simple bodies such as a semi-infinie body, a slab, a sphere, a finie or semi-infinie cylinder, he form of he funcion G(P, Q, ) is known Le us consider as an example he case of a finie cylinder, in which Ą will denoe he lower boom, S 3 he upper boom and S 2 ^he laeral surface The fundamenal sysem is he same cylinder, hermally insulaed on S 1 and S 2 and kep a zero emperaure on S 3 In his fundamenal sysem, Green's funcion G(P, Q, ) can be obained relaively easily, as also he emperaure T Q (Q, f) The unknown funcion ipx(r x > i) on he surface S x will be deermined from (233) In he paricular case of mixed boundary condiions under consideraion, he soluion presens no major difficulies I should be observed in addiion ha he problem reaed here can be solved in a simpler way by direc inegraion of Eq (11), he inegral Eq (223) no being considered Much greaer difficulies are encounered for solving he nex problem, concerning he semi-infinie cylinder (Fig 2) wih mixed and disconinuous boundary condiions on he laeral surface FIG 2 Le he emperaure of he par S x of he laeral surface be zero, he par is 2 of ha surface being insulaed and le he emperaure of Ą be <p 3 (i? 9, i), R e >S 3 Disconinuiy of boundary condiions occurs on he regular laeral surface Sx+iSg "As a fundamenal sysem, we assume a semi-infinie cylinder hermally insulaed

9 872 W Nowacki over he enire area S x +S^ In his fundamenal sysem, we find easily he funcions G(P, Q> 0 arl d 7o(2> 0- The unknown funcion ip^r^ ) can be deermined from he inegral Eq (223) only However, an accurae soluion of his inegral equaion is conneced wih serious mahemaical difficulies, which can be overcome only in a few simple cases of saionary flow in paricular In more complex cases, we mus have recourse o approximae soluions of Eq (223) A similar case of disconinuous boundary condiions is ha of he semi-infinie cylinder of Fig 3 The deerminaion of Green's funcion G(P, Q, ) and T(Q, ) Jfn FIG 3 in he fundamenal sysem (hermal insulaion over Ą and S 2 and zero emperaure over 5 S ) is no difficul in his case eiher Difficulies are firs encounered in he soluion of he inegral Eq (223) Le us consider wo semi-infinie cylinders joined in he z = 0 plane A secional view of his sysem is represened by Fig 4 The surfaces Sj and S" are hermally FIG 4 insulaed; SI and SI 1 are kep a zero emperaure The surface S[ = SJ 1 is ha of join beween he regions T and u The fundamenal sysem is consiued by wo semi-infinie cylinders: he cylinder z, hermally insulaed on SJ + S 1! and kep a zero emperaure on Si and he cylinder u hermally insulaed on Sp+S 1?, and kep a zero emperaure on SJ Le us obain he funcions G^P, Q, i), Tl (Q, ), P,Qej

10 Mixed boundary-value problems in hea conducion 873 and G"(P, Q, ), T"(Q, i) in he region n, The emperaure gradien in he plane will be assumed o be unknown a a We wrie Eq (217) firs for he region I and hen for he region n Leing he poins Q x e x and g u e u end o he poin R[ on SI = S" and making use of he condiion of idenical emperaure on SI, [(pkr^ ) = <p (Ri> OL we obain he required inegral equaion for he unknown funcion ip^rj, *) The soluion of (11) wih he boundary condiions (13) leads o ha of he inegral Eq (223) In more complex cases wih mixed condiions on he surface S(S= S 1 -\- S 2 +,S k ), a se of inegral equaions will be obained FIG 5 Le us consider he solid body represened in Fig 5 in which hermal sources ac, and mixed boundary condiions are prescribed on he surface 5 1 = (234) T(Ri, ) = -PiCRi, 0 ĄsĄ; ~Jp"^ = %(* 2, 0 T(i?, 0 = <p 8 (i? 8,0 ĄeĄ; T(R, ) = Le us assume as a fundamenal sysem he same body, hermally insulaed on he surfaces S l S 2 and S 3, and kep a zero emperaure on 5 d In his fundamenal se, Green's funcion G(P, Q, ) and emperaure T 0 {Q, ) mus be found The laer will be obained as a soluion of (11) wih he boundary condiions (234) from he equaion (235) T(Q, ) = T 0 (Q, ) +«/ dv / J W 1 (R 1, r)g(r 1,Q,- r)ds Rl dx JJrp 3 (R 3, r)g(r 3,Q,-r)dS Ri s,

11 874 W Nowacki Leing he poin Q end firs o R x e S l and hen o Q, R 3 e S 3, and aking he firs and he hird of he condiions (234), we obain a se of wo inegral equaions (236) cpik ) - T 0 (Ri )+xfdrj} Vl(R l r)g(r l R[, - r)ds Rl o s x + xjdrjj ip 3 (R s, T) G(R S, R» - T)dS Ri> R l R[ e Ą s cp 3 (R' 3, ) = T 0 (R^ ) + xjdr}j ^(R,, r)g(r v Rś, - r)ds Rl 0 S + xjdr Jjy s (R 3, r)g(i? 3, R' 3, - x)ds R3, R 3> R' 3 es 5 o s' a On solving his for y>i(ri, ) and ip 3 (_R 3, f), we find he emperaure T(Q, ) from Eq (235) Le us reurn o he iniial problem of deermining he emperaure in he body of Fig 1, The boundary condiions will he somewha modified, i being assumed ha he hea exchange over he surface S is free ar( (237) f lł ^ + ht(r l ) =0, h= cons, ĄeĄ on y deermining in he fundamenal sysem assumed he Green's funcion [Eqs (21) and (22)], and he emperaure T o [Eqs (212) and (213)], we obain Eqs, (216) and (217) wih y>i(^i» 0 replaced by ht(r u ) In his way, (217) akes he form i (238) T(Q, i) - T 0 (Q, )-xh] dr JJ T(R U r) G^, Q,-r) ds Rl o s, Leing now he poin Q end o R[ on S u we obain an inegral equaion of he second kind i (239) T(R[ ) = T 0 (R 1, i) - hx J dx J T(R U r)g(r u R[, - r)ds Rl 0 S Having deermined T(R 1} ) on S from (239), we find emperaure T(Q, ) from (238) FIG 6 Le us proceed now o solve he wo-dimensional problem Le us consider he infinie cylinder wih cross-secion S Le he conour of his cross-secion be composed of secionally regular arcs s l s 2, s s (Fig 6) Le us assume ha he emperaure

12 Mixed boundary-value problems in hea conducion 875 field is independen of he variables x z (he x g axis is parallel o he axis of he cylinder) Le he emperaure field in he cylinder be produced by hea sources W(P, ), and by surface heaing The emperaure in he region S is deermined by equaion if *-~ \Xiy X 2 ) Jj (240) xv\t{p, ) f(p, ) = M(P, ), wih he boundary condiion (241) T(P, 0) = 0, and he boundary condiions T(R l ) = 99^, ), R x es x, 1 dx 2 ^ dy* ' (242) 8T( f* f) - y>^r b ), R 2 e s,, Le us deermine Green's funcion G(P, Q, f) from he hea equaion (243) xv?g{p, Q, f)-g{p,q,) = ~d(p~q)5(i), P,QeS, wih he iniial condiion (244) G(P, Q, 0) = 0, and he boundary condiions <245) dg{r 2, ) 3n = ' * 26S2 ' 8n G(R 3,)=0, i? Making use of Green's formula in he plane S, we obain equaions r r Q es <246) T(Q, ) = T 0 (Q, ) + x I dx I yi^ru -r)g(i? 1; Q,-x)ds^_,, 0 j, -^ 5l where (247) TJQ, 0=1 f/(p)g(p, g, 0^+ I dx \ \ M(P, x)g(p, Q, J J J J J s c c -\- % \ dx \ ipzirz, T)G(-R 2 > 2> ~' J J,-*/dx 0 s, P,QeS, R x es x, R z i

13 876 W Nowacki The unknown funcion ip^r^ i) will be found from equaion (248) <p x {R[,)=T 0 (K,) + xfdrj Wl{R,, r)g(r 1,R[,-r)ds x, o», which will be obained from (246) by leing he poin Q end o R[ on he boundary s^ Similarly, he previous resuls, obained for periodic and seady emperaure, can be generalized o wo-dimensional problems of hea conducion 3 Second Soluion Mehod We shall give anoher varian of he soluion mehod of (11) wih mixed boundary condiions (13) The difference in he procedure will consis in a differen choice of he fundamenal sysem While in he previous case hermal insulaion exended over S x and Ą for he funcion G, now, for he funcion G*, we shall ake he same body wih he surfaces S ± and 5 a kep a zero emperaure The funcion G*(P, Q, } should herefore saisfy he hea equaion (31) «V»G*(P,fi, )-G*(P, Q, 0 = -d(p-q)d(), wih he iniial condiion (32) G*(P,G, 0) = 0, and he boundary condiions G*(R u Q,) = 0, R 1 es 1 ; (33) G*(R i,q,) = 0, R 2 es 2 ; G*(R a,q,) = 0, ReS 3 If now Greenes formula (29) is applied o he funcions Tand G*, and if he boundary condiions (13) and (33) are considered, we obain (34) f(q, p) = Jfffip) G*(P, Q, p)d P + fjj M(P, p) G*(P, Q, p)d P or (3-5) s, Le us observe ha he quaniies involved in TJ (Q, p) are known The funcion T*(Q, 0 can be obained by solving he hea equaion (36) * V 2 T*(P, 0 - f*(p, ) = -M(P, ), Pe,

14 Mixed boundary-value problems in hea conducion 877 wih he iniial condiion T*(P, 0) =f{p) and he boundary condiion (3 ' },0 = 0, ĄeĄ; T O *(R S, i) = ^s(i? 3,0, R 3 es 9 On performing on (31), (33), (36), (37) one-sided Laplace ransformaion, we obain from Green's formula an equaion, which, afer inverse ransformaion, akes he form (38) T*(Q, ) = fjjf(p)g*(p, Q, )d P + J dx j(j M(P, r) G*(P, Q> -r)d F On performing now he inverse ransformaion on (35), we obain (39) 2m 0 = T*(Q, )-xfdtjf<p z (R r) dg *( R *> 8 Ck±zA d s Ri o s I is known, however, ha he boundary condiion assigned on S z is (310) d^'^vifofl, ĄeĄ Le us make use of his condiion by leing he poin Qei) pass o he curren poin R' 2 E S 2, and by performing he equaion d/dri Thus, from (39), we find j R,, K ~ r ) ram f» fw ^fw0 j 0 S«where d/dn' denoes he normal derivaive a R' z e S z The inegral Eq (311) will be used o rind he unknown funcion (p 2 (R 2, ) From (39) we can find he emperaure am 0- Le us consider also Green's funcion K*(P, R' 2, ), saisfying, in our fundamenal sysem, he hea equaion (312) H V 2 K*(P, R'i, ) - K*(P, R' 2, 0 = 0, P e, wih he iniial condiion K*(P, R' 2, 0) = 0, and he boundary condiions (313) K*(R l 2& 0 = 0, RjeS!, ' where JdrfJ 5(R i -R' 2

15 7 - Nowacki y applying Green's formula o he funcions G* and K* on which one-sided Laplace ransformaion has been performed, we find ha (314) j hence, on performing he inverse Laplace ransformaion, we obain earing in mind (315), Eq (36) can be represened in he form J ax J J (316) y> 2 (i? 2, 0 = ^ h b s Similarly o Sec 2, we can easily pass from Eqs (39)-(310) o hose for harmonic emperaure or seady-sae flow Le us consider also he problem wih he following boundary condiions on he surface S 1 of he body (317) ar( f 2> /} + ArCUa, 0 = 0, on Making use of he inegral Eq (316) and bearing in mind ha o and aking ino consideraion he second of he condiions (37), we find (318) ht{r 2, ) = ^T 1- J ax J J T^, T) ^ rf oj J J o s % Thus in he case of he boundary condiions (317), an equaion of he second kind is obained 4 Soluion for a ody wih Slis and Insulaing Diaphragms Le us consider a simly conneced body bounded by he surface S and having a sli, of which he (upper and lower) surfaces are denoed by S% and S' 2 ', respecively he remaining par of he surface being denoed by S v Le he body be subjec o inernal hea sources and surface heaing The emperaure T(P, i) mus saisfy he hea equaion (41) x V 2 TCP, ) - f(p, ) = -M(P, ), Pe, wih he iniial condiion ( 4-2) T(P,0)=f(P),

16 Mixed boundary-value problems in hea conducion 879 and he boundary condiions (4l3) 2X1*1,0-0 on Ą, R 1 es 1, r(u0 = 0 on S^iS^' Le us consider he funcion G(P, Q, f) in he basic sysem consiued by he same body Si wih no sli The funcion G(P, Q, ) mus saisfy he hea equaion (44), wih he iniial condiion (45) G(P, Q, 0) = 0 and he boundary condiion (46) G(R 1> Q,)=0 on Ą Le us perform on (41)-(46) he one-sided Laplace ransformaion, and apply Green's formula (47), Afer some simple rearrangemen, we obain S (48) 2XG P) = j ff f ( F W p, Q> P) d v + /// M(P,p) G(P, Q, p)d p or SŚ+S2' ' ~dn 2 (49) T(Q, p) = T 0 (Q, p) + x \ \ G(R 2, Q, p)\ -^ where + denoes he upper and he lower par of he surface S 2 The funcion G having been defined in he region wih no sli, his funcion is differen from zero on 5 a The emperaure gradiens on he surfaces S^ and S'^' are also differen from zero (he second boundary condiion of he group (43) being ha of zero emperaure) The expression in brackes under he inegraion sign in (49) can be reaed as an unknown funcion ip(r 2, ) Thus Eq (49) akes, on performing he inverse Laplace ransformaion, he form i (410) f(q, i) = r o (fi, 0 + «/ jj o s, (i?, r) G(R, Q, ) V 2 2 ds, Rl where (411) T 0 (Q, ) = JJJf(P)G(P, Q, - x)d +fdr jjj M(P, r) G{P, Q, - x)d ' 0 J Arch Mech Sos 2

17 880 W Nowacki Le us make he poin Q end o i?ś on Ą earing in mind ha (in agreemen wih he second boundary condiion (43) we have T(R i ) = 0 on S 2, we obain he following inegral equaion (412), r)g(r 2, JC, - x)ds Rl s y solving (412) we find he funcion ip(r 2, T) from which he emperaure field according o (410) can be found 1 FIG 7 Le us consider in urn he solid of Fig 7, subjec also o a emperaure field bu wih differen condiions Le he emperaure #(P, ) saisfy he hea equaion (413) H V 2 #(P, 0 - &(P, ) = - M{P, f), Pe, wih he iniial condiion f(p, 0) = 0 and he boundary condiions (414), ) 8n!, 0 = 0 on 5j, = 0 on Ą = We have assumed here ha he surface 5 2 is hermally insulaed y applying Green's funcion G(P, Q, ) as deermined by Eq (44) wih he iniial condiion (45) and he boundary condiion (43), we obain, on applying Green's formula (47) for he following relaion beween he funcion & and G (415) where, 0 = (T(R 2, = JJjf(P)G(P, Q, )d+jdrf}{m(p, r)g(p, Q,-r)d 1 A similar soluion mehod was applied in Ref [2] o he problem of deflecion of a membrane and orsion of a bar and in Ref [3] o ha of orsion of a bar ha is, o he soluion of Poisson's equaion in a plane

18 Mixed boundary-value problems in hea conducion 881 Denoing by <p(i? a, ) he unknown emperaure funcion on 5" a ' and S' z \ le us rewrie Eq (415) in he form (416) 0(fi, ) = 0 o (fi, 0 + J dx fj o Le us apply he condiion - - = 0 on S 2 y leing Q pass o R' 2 es 2> we obain he following inegral equaion of he firs kind ( ) O S, The funcion <p(i? 2 T) being now deermined, we can find he emperaure &(Q, ) from Eq (416) Le us consider he case in which he following condiions are required o be saisfied for he emperaure 6(P, 0) =f(p), in addiion o he iniial condiion 6(R 1,) = 0 on Ą, (418) 0(-R 2,0 = 0 on 5j, ^ = 0 on Si' 8n Making use of Green's funcion G(P, Q, ) (44) and he condiions (45) and (46), we obain for he emperaure d(q, ) he formula (419) 6(Q, i) = 0 o (fi, ) + nfdx fj ^I lil> G^, Q, - si where o flo(fi 0 = ////( p ) G ( p > 2, 0d + «/ dx Jfj M(P, x)g{p, Q,-r)d O Le us consider now he boundary condiions (418) Leing he poin Q e in (419) end firs o GJ on SŹ, and hen o Cj' on Sj' we find he se of equaions J si' (420) 0(q, 0 = 0 = 0 o (Cl,)+xjdrfj 0j0 06{R^ T) G(2?,', C S ', f- r)ds' Rz C, C Caro" ds

19 882 W Nowacki ćri 8n I «f AT f f " 0 Si These are inegral equaions of he firs kind wih unknown funcions a on A and <?(i?2'> 0 o n "S 1^'- On solving hem for hese unknown funcions we can deermine he emperaure from (419) FIG 8 Le us consider now he simply conneced body of Fig 8 wih an insulaing diaphragm S z inside he body Le he body be aced on by some hea sources inside he body which is also heaed on he surface Ą The diaphragm condiion is dt/dn, here being no hea flow across S z The emperaure should saisfy he hea equaion (422) K V*T(P, ) ~ T(P, ) = - M(P, ), Pe, wih he iniial condiion (4-23) T(P, 0) = 0 and he boundary condiions (424) T(R 1,) = 0 on S x, ReS,, a u u u "2; "2 "a ~r >->i > -K2 >->2 Assuming in he fundamenal se Green's funcion G(P, Q, ), saisfying he equaion (44) wih he iniial condiion (45) and he boundary condiions (46), and, finally, applying Green's formula (47) o he funcions G and T (obained by solving (422), we obain (425) T(Q,p) = T 0 {Q,p) - «[f [T + (R s,p) + T_(R 2,p);

20 Mixed boundary-value problems in hea conducion 883 where T 0 (Q,p) is given by (411) and T+ T_ de oe he emperaure on he upper and lower surface of he insulaing diaphragm, respecively These emperaures are unknown funcions Their sum is denoed by q} 2 (R 2, ) Thus, (425) akes he form (426) T(Q, p) = f o (Q, />)-*//?,(*, P) J G( - R *> n &P> ds Rx Leing he poin Qe end o R' & on S 2 and performing he operaion d/dn' on (426), we obain The lef-hand side of (427) represens he normal hea flow a R' z e S 2 However, in view of he hermal insulaion on Ą, his flow is zero Eq (427) is an inegral equaion of he firs kind, from which he unknown funcion <p 2 (i? 2, p) can be deermined, hus enabling us o find T(Q,p) from (426) On performing he inverse Laplace ransformaion of (426), we obain (428) T{Q, ) = T 0 (Q, )-xf dr jf <p(r z, o S The procedure jus described can be generalized o he case of exisence in he body of more han one insulaing diaphragm In he case of r diaphragms, we obain a se of T inegral equaions References [1] W NOWACKI, A boundary problem of hea conducion ull l'acad Polon Sci Cl IV V V, No 4, 1957 [2] W NOWACKI, O pewnym przypadku skręcania pręów [On cerain cases of orsion of bars], Arch Mech Sos, I, 5 (1953) [3] J ALLAS, On he orsion of a cylindrical bar wih slis, Koninkl Nederl Acad van Weenschappen, Proceedings, Series, 64, No 1,1961 [4] H S CARSLAW, J C JAEGER, Conducion of hea in solids, Oxford 1947 [5] R SETH, Mixed boundary-value problems, Calcua Mah Soc Golden Jubilee Commemoraion V (1958/59), par 1, 79-86, Calcua Mah Soc, Calcua 1963 [6] W NOWACKI, Mixed boundary problems in hea conducion, ull l'acad Polon Sci, Serie Sci Techn, 5, 11, (1963) Sreszczenie MIESZANE WARUNKI RZEGOWE W ZAGADNIENIACH PRZEWODNICTWA CIEPLNEGO Nawiązując do swej dawniejszej pracy [1] auor rozszerza podaną am meodę rozwiązy\ unia zagadnień przewodnicwa cieplnego w ciele sałym z mieszanymi warunkami brzegowymi na nieusalone przepływy ciepła Podano dwa wariany rozwiązania, przy użyciu dwu różnych ak zwanych

21 884 W Nowacki układów podsawowych W obu warianach sprowadza się rozwiązanie zagadnienia do rozwiązania równania całkowego pierwszego rodzaju W końcu omówiono ok posępowania dla przypadku isnienia w ciele sałym szczelin oraz przesłon izolacyjnych P e 3 K) M e CMEIUAHHIE KPAEŁIE YCJIOHH SAJOA^AX TEnJIOnPOOJJHOCTH CH3H c oflhoił H3 npenbiflymix pa6ot [1] atop pacniipjiet npheflehhbiił Hef penehhh onpoco, KacaiomnxcH TenjionpoOflHOCTH Tepfloro Tejia c pa3pbihbimh KpaebiMH ycjiohhmh fljih HecTaqaoHapHLix notoko Tenna JJaioTCH fla aphahta penrehhji npn Hcnon- 3OannH flyx pasjni'qhbix, Tax Ha3biaeMbix, OCHOHIX CHCTeiw OOHX apaahtax peiilehhe 3aflaiH CO^HTCH K peiflehhio HHTerpanbHoro ypahenhh neporo pofla 3aKjno^ieHHe o6cy>kflanpoiiecc penlehhh nph HaroraHH TepflOM Tejie iqejieii H H30Ji^n0HHbix DEPARTMENT OF MECHANICS O CONTINUOUS MEDIA ITP POLISH ACADEMY OF SCIENCES Received January 9, 1964